Solving 3D Geometric Constraints for Assembly Modelling

Transcription

1 Int J Adv Manuf Technol () 6: Springer-Verlag London Limited Solving 3D Geometric Constraints for Assembly Modelling J. Kim, K. Kim, K. Choi and J. Y. Lee 3 School of Mechanical and Industrial Engineering, Pohang University of Science and Technology, Pohang South Korea; Production Engineering Research Center, LG Electronics, Pyungtaek South Korea; and 3 Computer Software Technology Lab, Electronics and Telecommunications Research Institute, Taejon South Korea A geometric constraint solving method is presented that takes well-constrained mating conditions between a base and a mating component and directly transforms them into a 4 4 matri that determines the relative orientation and location of the mating component with respect to the base component. In the proposed procedure, the 4 4 transformation matri is determined by directly computing a rotation matri T R and a translation matri T L that define the relative orientation and location of the mating component, respectively. Thus, first, the rotation matri is computed by solving a set of linear constraint equations associated with the orientation of two mating components. After repositioning the mating component by applying the rotation matri T R, the translation matri is calculated by solving a set of linear constraint equations associated with location. This new method is computationally very effective, since the transformation matri for relative orientation and location of the mating component is algebraically derived directly from the linear equations associated with the mating conditions. Keywords: Assembly modelling; Constraints solving; Geometric constraints; Mating conditions. Introduction Assembly design is an important part of the product design activity, since most manufactured products are assemblies of components. It is well known that the assembly design has a significant impact on many downstream activities such as process planning, production planning and control, and packaging. Assembly design involves the creation of assembly models that specify the relative location and orientation of components. In the design activity, component geometry is assembled together to create an assembly model. Surface mating constraints are used to locate and orient components with respect Correspondence and offprint requests to: Dr K. S. Kim, Pohang University of Science and Technology, School of Mechanical and Industrial Engineering, San 3, Hyoja-dong, Pohang , South Korea. kskimpostech.ac.kr to each other. Supporting the design activity requires the development of constraint solving methodologies that take the surface mating constraints and transform them into 44 transformation matrices. A 4 4 matri determines the relative location and orientation of a mating component with respect to a base component. By repositioning the mating component using the transformation matri, the design intent can be realised. Thus, it is essential to develop an efficient constraintsolving method for assembly modelling [,]. A number of papers have described the method of deriving the transformation matrices from a set of assembly mating relationships. An approach based on numerical procedures, in which the mating constraints are typically represented by a set of nonlinear equations and solved simultaneously by using a modified Newton Raphson iteration procedure, is described in [3 5]. This approach, based on numerical computing procedures, has some inherent disadvantages: it does not guarantee a solution each time and the final solution is often dependent on the initial values. Another approach [6 ] is based on algebraic procedures, in which each mating operation is subdivided into a sequence of rotation and translation operations by using DOF (degrees of freedom) reduction. Thus, the transformation matri for determining the relative orientation and location of a mating component with respect to a base component is written as a product of rotation and translation matrices. In this approach, the set of rotation and translation operations is typically determined by sequential DOF reduction using a look-up-table procedure. The current DOF and the net mating constraint to be applied are inputs, and the resulting DOF and rotation and translation operations to satisfy the mating constraint are the output. The table has approimately entries. In this paper, we present a geometric constraint-solving method that takes well-constrained assembly mating conditions between a base and a mating component and directly transforms them into a 4 4 matri that determines the relative orientation and location of the mating component with respect to the base component. In the proposed procedure, the 4 4 transformation matri is determined by directly computing a rotation matri T R and a translation matri T L that define the relative orientation and location of the mating component, respectively. Thus, we first compute the rotation matri by solving a set of linear constraint equations associated with the orientation of two

2 844 J. Kim et al. mating components. After repositioning the mating component by applying the rotation matri T R, we calculate the translation matri by solving a set of linear constraint equations associated with location. The remainder of this paper is organised as follows. Section presents an overview of the assembly mating conditions. Section 3 describes the proposed constraint-solving method for assembly modelling. Section 4 presents an eample to show the effectiveness of the proposed method. Section 5 shows the implementation results. Section 6 concludes with remarks.. Mating Conditions in the Assembly Modelling There are three major types of the mating conditions:. Distance: specifies the distance between the mated components.. Angle: specifies the angle between mated components. 3. Alignment: specifies whether the component is aligned on the same side or opposite side of the plane. The mating conditions considered in this paper are alignment type constraints such as against, fits, and coplanar. A set of well-constrained mating conditions uniquely defines the 4 4 transformation matri which determines the relative orientation and location of the mating component. More detailed description of these mating conditions are given below. In the equations below, the superscripts b, m, mr, and ma indicate the base component, the mating component, the mating component after rotation, and the mating component after assembly, respectively.. Against The against condition holds between two planar faces and requires the two faces to touch each other (Fig. ). The designated faces, shaded in Fig., are the faces to be mated. Each face is specified by its unit normal vector N and any one point P on the face in terms of its local coordinate system. This condition is accomplished by constraining the two normal vectors to be opposite to each other, and the two points to lie on the same plane at which the two faces mate. Thus, these constraints can be epressed as: Nb Nma N b = N b y = N ma y = Nma () N b N ma Pb P ma [N b N b y N b ] P b y P ma y P b y P ma = () Equations () and () are associated with the relative rotation and translation of the mating component with respect to the base component, respectively.. Fits The fits condition holds between two cylindrical faces: a shaft face, and a hole face, as shown in Fig.. The fits condition is accomplished by requiring the centre aes of shaft and hole components to be parallel and a point P m on the ais of the mating component lies on the ais of the base component. An ais is defined by a unit direction vector and a point on it. The hole ais is specified by a point P b and a unit direction vector N b defined in terms of its local coordinate system. Similarly, the shaft ais is specified by a point P m and a unit direction vector N m in terms of its local coordinates system. Thus, the constraint equations for fits conditions can be written as: N b = Nma P ma N b P b N ma for aligned fits conditions for anti-aligned fits conditions = Pma y N b y P b y = Pma N b P b Equations (3) and (4) are associated with the relative rotation and translation of the mating component with respect to the (3) (4) Fig.. Against condition. Fig.. Fits condition.

3 Solving 3D Geometric Constraints 845 Fig. 3. Coplanar condition. base component, respectively. Equation (4) yields three combinations of equations for each fits condition. In general, one of these equations is redundant because the relation yields only two independent equations instead of three. However, it is necessary to keep all three to cover the case where the centreline of a component is parallel to any of the base coordinate aes so that the relation gives only two equations..3 Coplanar The coplanar condition is assigned between two planar faces when they lie in the same plane, as shown in Fig. 3. It is similar to the against constraint ecept that the two normal vectors N b and N m are required to be in the same direction. Thus, these constraints can be epressed by: Nb Nma N b = N b = y N ma = y Nma (5) N b N ma Pb P ma [N b N b y N b ] P b y P ma y P b y P ma = (6) 3. Solving 3D Assembly Constraints The relative orientation and location of the mating component with respect to the base component is represented by a 4 4 transformation matri. The transformation matri can be written as T = R R R3 L R y R y R 3y L y R R R 3 L = R L (7) Fig. 4. Mating conditions for assembly modelling. As shown in equations (7), the transformation matri T can be determined by computing two submatrices: a 3 3 rotational submatri R and a 3 translational submatri L. The transformation matri T can be represented by the product of a translation matri T L and a rotation matri T R as shown below: L L y T = T L T R = L R R R3 R y R y R 3y R R R 3 These matrices are determined sequentially in the proposed constraint solving procedure. In the new method, we first derive T R from the rotational relationships between the mating components, and then derive T L from the translational relationships between the base component and the mating component rotated after repositioning the mating component by applying the rotation matri T R. 3. Computing the Rotation Submatri R As stated earlier, the against, fits and coplanar conditions are used to constrain rotation of components in assembly modelling. Each of these mating conditions has associated with it a pair of direction vectors that are either aligned (when the angle between the directions is ) or anti-aligned (when the angle between the directions is 8 ). The associated direction vectors are the unit normal vectors of the base and mating faces for the against and coplanar constraints, and the ais direction vectors of the cylinder and hole components for the fits constraint. When we are given two independent pairs of direction vectors, (N b,n m ) and (N b,n m ), from the well-constrained mating conditions between a base component and a mating component, as shown in Fig. 4, the equation associated with rotation of components are epressed as = RN m = RN m where i = Nb i for aligned constraints (i =,) N bi for anti-aligned constraints (i =,)

4 846 J. Kim et al. Fig. 5. Assembly components before and after assembly. Here, i is a mating direction vector after the mating component is repositioned by applying the rotation matri T R. When N m 3 and 3 are defined as N m 3 = N m N m 3 = then, the relation between N m 3 and 3 is derived as 3 = RN m 3 These equations are rewritten as [ 3 ] = R[N m N m N m 3 ] Thus, the rotational submatri R is obtained by R = [ 3 ][N m N m N m 3 ] 3. Finding the Translation Submatri L The translation submatri L is computed algebraically by solving the mating constraints associated with translation after repositioning the mating component by applying the rotation matri T R. After repositioning, the mating direction vectors are parallel to their corresponding base direction vectors and a point on the mating component after assembly, P ma, is epressed as P ma = P mr + L where P mr is a point on the mating component after repositioning and is obtained by P mr = RP m. Thus, the constraint equations associated with the translation of mating components are epressed as follows. Here, the translation submatri L is determined by algebraically solving a set of three independent translational constraint equations.. against and coplanar conditions Since the normal vectors of the mating components are parallel after repositioning, the against or coplanar condition requires that one point on the mating face of the mate component lies on the mating face of the base component. Thus, the translational constraint equation for the against or coplanar condition is epressed as follows: Pb (P mr + L ) ) [N b N b y N b ] P b y (P mr y + L y ) = P b (P mr + L. fits constraint Since the centre aes of the mating components are parallel after repositioning, the fits constraint requires that one point P m on the ais of the mating component lies on the ais of the base component. Thus, the translational constraint equations for the fits condition are epressed as follows: (P mr + L ) P b N b 4. Eamples = (Pmr y + L y ) P b y N b y = (Pmr + L ) P b N b The following eample illustrates the effectiveness of the proposed constraint-solving procedure eplained in this paper. Consider the two assembly components C and C, shown in Fig. 5. Each component is defined in its local coordinate system. The base component C is a chamfered block with a blind hole and the mate component C is a wedge with a cylindrical shaft. The two components are constrained by a coplanar and a fits condition. The

5 Solving 3D Geometric Constraints 847 coplanar condition holds between a base face F b and a mate face F m. Each planar face is defined by a unit normal and a point on it. The fits condition holds between the blind hole of C and the cylindrical shaft of C.Aholeor shaft ais is also defined by a unit direction and a point on it. Table summarises the geometric data for computing the 4 4 transformation matri.. Computing the rotation submatri R When the two direction vectors N m and N m for the mate component are given, the third direction vector can be computed by N m 3 = N m N m = Then, the rotation submatri R is epressed as follows: R = [ 3 ][N m N m N m 3 ] where = N b = from the coplanar condition = N b = the anti-aligned fits condition from Table. Geometric data for computing the transformation matri. Coplanar Fits Base component (C ) Mating component (C ) N b =, Pb = 6 6 N b =, Pb = Nm =, Pm = 3 =, Pm = 3 3 Nm 3 3 = = Thus, the rotation submatri R is computed as R = = =. Finding the translation submatri L Using the rotation submatri R obtained in the previous step, two points P mr and P mr on the repositioned mating component are computed first by P mr = RP m = 3 3 and P mr = RP m = 3 3 Then, the translational constraint equations are derived from the mating conditions as follows: From the coplanar condition, Pb (P mr + L ) ) [N b N b y N b ] P b y (P mr y + L y ) = P b (P mr + L 6 ( + L ) ) or [ ] 6 (3 + L y ) = ( + L From the fits condition, (P mr + L ) P b = (Pmr y + L y ) P b y = (Pmr + L ) P b N b N b y N b

6 848 J. Kim et al. Fig. 6. Assembly components before and after assembly operations. or ( 3 +L ) = (3 + L y) = ( 3 +L ) Using these equations, the translation submatri L is computed as L = Thus, the transformation matri T for relative orientation and location of the mating component with respect to the base component is obtained as T = 5. Implementation An assembly modelling system based on the proposed approach has been implemented in an SGI workstation. The ACIS solid modeller is used as the geometric modelling kernel in the assembly modelling system. Figure 6 shows two sets of assembly components before and after assembly operations. 6. Conclusions In this paper, we present a geometric constraint-solving method that takes well-constrained assembly mating conditions between a base and a mating component and directly transforms them into a 4 4 matri that determines the relative orientation and location of the mating component with respect to the base component. In the proposed procedure, the 4 4 transformation matri is determined by directly computing a rotation matri T R and a translation matri T L that defines the relative orientation and location of the mating component, respectively. Thus, we first compute the rotation matri by solving a set of linear constraint equations associated with orientation of two mating components. After repositioning the mating component by applying the rotation matri T R, then, we calculate the translation matri by solving a set of linear constraint equations associated with location. This new method is computationally more effective than the current numerical methods based on a modified Newton Raphson iteration procedure or algebraic methods based on a sequential DOF reduction procedure, since the transformation matri for relative orientation and location of the mating component is algebraically derived directly from

7 Solving 3D Geometric Constraints 849 the linear constraint equations associated with the assembly mating conditions in the proposed method. Acknowledgement This work was supported in part by ETRI and BK. References. R. Sodhi and J. U. Turner, Towards modeling of assemblies for product design, Computer-Aided Design, 6(), pp , U. Roy, P. Banerjee and C. R. Liu, Design of an automated assembly environment, Computer-Aided Design, (9), pp , A. P. Ambler and R. J. Popplestone, Inferring the position of bodies from specified spatial relationships, Artificial Intelligence, 6, pp , D. Rocheleau and K. Lee, System for interactive assembly modeling, Computer-Aided Design, 9(), pp. 65 7, F. Tomas and C. Torras, A group-theoretic approach to the computation of symbolic part relations, IEEE Transactions on Robotics and Automation, 4(6), pp , L. S. Haynes and G. H. Morris, A formal approach to specifying assembly operations, International Journal of Machine Tools and Manufacture, 8(3), pp. 8 98, G. A. Kramer, Solving geometric constraint system: A case study in kinematics, MIT Press, J. U. Turner, S. Subramaniam and S. Gupta, Constraint representation and reduction in assembly modeling and analysis, IEEE Transactions on Robotics and Automation, 8(6), pp , R. Anantha, G. A. Kramer and R. H. Crawford, Assembly modeling by geometric constraint satisfaction, Computer-Aided Design, 8(9), pp. 77 7, V. N. Rajan, K. W. Lyons and R Sreerangam, Generation of component degrees of freedom from assembly surface mating constraints, Proceedings of ASME Design Engineering Technical Conference, DETC97/DTM-3894, September 997.